Two to the power of two to the power of...

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Let

\[S=\dfrac{1}{2+1}+\dfrac{2}{2^2+1}+\dfrac{4}{2^4+1}+\dfrac{8}{2^8+1}+\dots+\dfrac{2^{2014}}{2^{2^{2014}}+1}\]

The exact value of \(S\) can be expressed in the form \(S=\frac{1}{a}-\frac{b}{2^b-c}\), where \(a\), \(b\) and \(c\) are real numbers. What are the last 3 digits of \(\log_{2}(abc)\)?

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